Fiscal policy and corruption

Abstract

This paper develops a simple model of corruption between a tax inspector and a taxpayer in which inspectors’ heterogeneity helps explain why it may be optimal for a revenue-maximizing government to tolerate corruption. In our model, tax inspectors could accept a bribe from a taxpayer for under-reporting his income or to avoid harassment in the form of red tape. The government’s problem is to design a tax policy and the associated monitoring probability to enforce it. Raising tax rates increases enforcement costs as it increases bribery incentives, so it is optimal not to set tax rates at a too high level. The optimal tax policy includes an optimal supervision level required to reduce corruption, which in turns depends on tax collectors’ capacity to impose red tape on taxpayers. When bureaucrats are heterogeneous in terms of their capacity to impose red tape costs, an intermediate detection probability might be appropriate efficiency-wise even if it means tolerating some level of corruption. Bureaucrats’ heterogeneity as the likely source of observation of corruption in equilibrium is generic to various environments in which agents’ delegated power is misused for private gains and various examples are discussed in the paper. We also show how a government could face lose–lose as well as win–win situations in the conduct of its revenue collection policies.

Appendix

Proof of proposition 1

Net tax revenues to the government are as follows:

$$\begin{aligned} RN\tau _1= & {} {\frac{p_1}{(1 - p_1)}} w - c_{1} - {\alpha p}_{1} \\ RN\tau _2= & {} (1 - \pi _{1}) \left( {\frac{p_2}{(1 - p_{2})}} w - c_2\right) - {\alpha p}_{2} \end{aligned}$$

We calculate the critical proportion of type 1 bureaucrats yielding higher net revenues under a flexible regime than under a no-corruption regime:

$$\begin{aligned}&RN\tau _1 < {RN\tau _2} \\&\quad \Leftrightarrow {\frac{p_{1} w}{( 1 - p_{1})}} - c_{1} - {\alpha p}_{1} < (1 - \pi _{1}) \left( {\frac{p_{2} w}{( 1 - p_{2})}} - c_{2} \right) - {\alpha p}_{2} \end{aligned}$$

where:

$$\begin{aligned} p_1= & {} 1 - \left( {\frac{w}{\alpha }}\right) ^{1/2}; \quad p_2 = 1 - \left( {\frac{(1 - \pi _{1})w}{\alpha }} \right) ^{1/2}; \\ {\frac{p_{1} w}{(1-p_{1})}}= & {} (\alpha w)^{1/2} - w; \quad {\frac{p_{2} w}{(1-p_{2})}} = \left( {\frac{\alpha w}{1-\pi _{1}}}\right) ^{1/2} - w \end{aligned}$$

Thus,

$$\begin{aligned}&({\alpha w})^{1/2} - w - c_{1} - \alpha + ({\alpha w})^{1/2} < (1-\pi _{1}) \left[ \left( {\frac{\alpha w}{1-\pi _{1}}}\right) ^{1/2} - w - c_{2} \right] \\&\qquad - \alpha + ((1-\pi _{1}) {\alpha w})^{1/2} \\&\quad \Leftrightarrow 2 ({\alpha w})^{1/2} - w - c_{1} < 2( (1-\pi _{1}) {\alpha w})^{1/2} - (1-\pi _{1})w - (1-\pi _{1}) c_{2} \\&\quad \Leftrightarrow (1-\pi _{1}) (w+c_{2}) - 2(1-\pi _{1})^{1/2} ({\alpha w})^{1/2} + 2({\alpha w})^{1/2} - w - c_{1} < 0 \end{aligned}$$

Substituting,

$$\begin{aligned} x = (1-\pi _{1})^{1/2} \quad \hbox {et}\quad x^2 = (1-\pi _{1}) \end{aligned}$$

We have:

$$\begin{aligned} x^2 (w+c_{2}) - 2x({\alpha w})^{1/2} + 2({\alpha w})^{1/2} - w - c_{1} < 0 \end{aligned}$$

Solutions to this inequality are:

$$\begin{aligned} x_{1}= & {} {\frac{ \sqrt{\alpha w} - \sqrt{\alpha w - 4( w + c_{2}) ( 2 \sqrt{\alpha w} - w - c_{1})}}{2(w + c_{2})}} \\ x_{2}= & {} {\frac{\sqrt{\alpha w} + \sqrt{\alpha w - 4( w + c_{2} )( 2 \sqrt{\alpha w} - w - c_{1})}}{2( w + c_{2})}} \end{aligned}$$

Hence, $(1 - x_1^2) > \pi _{1} > (1 - x_2^2)$ $\square$

Proof of Proposition 2

Social costs under the no-corruption regime $(SC_{sc})$ are:²⁶

$$\begin{aligned} SC_{sc} = \pi _{1} c_{1} + (1 - \pi _{1}) c_{2} + \alpha p_{1} \end{aligned}$$

While social costs under the flexible regime $(SC_{c})$ are:

$$\begin{aligned} SC_{c} = (1 - \pi _{1}) c_{2} + \alpha p_{2} \end{aligned}$$

Comparing these costs, we have:

$$\begin{aligned}&SC_{sc} > SC_{c} \\&\quad \Leftrightarrow \pi _{1} c_{1} + (1 - \pi _{1}) c_{2} + \alpha p_{1} > (1-\pi _{1}) c_{2} + \alpha p_{2} \\&\quad \Leftrightarrow \pi _{1} c_{1} + \alpha p_{1} > \alpha p_{2} \end{aligned}$$

This inequality always holds since $p_{1} > p_{2}$. Social costs are thus lower under a policy that allows some level of corruption then under a fiscal regime that completely eliminates it.²⁷

Let us turn now to the costs imposed on firms $(FC_{sc})$ under these two fiscal regimes. Under the no-corruption regime, firm costs are:

$$\begin{aligned} FC_{sc} = \pi _{1} (T_{1} + c_{1}) + (1 - \pi _{1}) ( T_{1} + c_{2}) \end{aligned}$$

Costs imposed on firms $(FC_{c})$ under the flexible regime are:

$$\begin{aligned} FC_c = \pi _{1} B + (1 - \pi _{1} ) (T_{2} + c_{2}) \quad \hbox {where} \quad B = (T_{2} + c_{1}) \end{aligned}$$

Comparing these costs, we have:

$$\begin{aligned} FC_{sc} > FC_{c} \end{aligned}$$

If we compare the first term of each equation, we note that:

$$\begin{aligned} \pi _{1} (T_{1} + c_{1}) < \pi _{1} (T_{2} + c_{1}) \end{aligned}$$

While for the second term, we have:

$$\begin{aligned} (1 - \pi _{1}) ( T_{1} + c_{2} ) < ( 1 - \pi _{1} ) (T_{2} + c_{2}) \end{aligned}$$

This implies that the costs imposed on firms are higher under the flexible regime. Hence, while the flexible regime yields higher net tax revenues and lower social costs than the no-corruption regime, we see that the costs imposed on firms are greater under the flexible regime than under the no-corruption regime. Under the flexible regime, corrupt firms pay bribes equal to $T_{2}+c_{1}$ to type 1 bureaucrats, while honest firms pay higher tax transfers (i.e. $\tau _{2} > \tau _{1}$). $\square$

Proof of Equation (7)

We have:

$$\begin{aligned} \max _B&(T + c - B)^\eta \left( B - {\frac{pw}{(1 - p)}}\right) ^{1-\eta } \\ \hbox {s.t : }&B \ge {\frac{p}{(1 - p)}} w \\&B \le T + c \end{aligned}$$

From first order conditions, we get:

$$\begin{aligned}&-\eta (T+c-B)^{\eta -1} \left( B - \left( {\frac{pw}{(1-p)}} \right) \right) ^{1-\eta } \\&\quad \quad + (T+c-B)^{\eta } (1-\eta ) \left( B - \left( {\frac{pw}{(1-p)}} \right) \right) ^{-\eta } = 0 \\&\quad \Leftrightarrow \eta ( T+c-B)^{\eta -1} \left( B - \left( {\frac{pw}{(1-p)}} \right) \right) ^{1-\eta } \\&\quad = (T+c-B)^{\eta } (1-\eta ) \left( B - \left( {\frac{pw}{(1-p)}} \right) \right) ^{-\eta } \\&\quad \Leftrightarrow \eta \left( B - {\frac{pw}{(1-p)}} \right) = (1-\eta ) (T+c-B) \end{aligned}$$

which yields the following relationship:

$$\begin{aligned} B = \eta {\frac{pw}{(1 - p)}} + (1 - \eta ) (T + c) \end{aligned}$$

$\square$

Corollary 3

Derivation of critical bargainning power $\eta ^{*}$.

Proof

To obtain firms’ costs that are lower when a government chooses a flexible fiscal regime, we need:

$$\begin{aligned}&FC_{sc} > FC_c \\&\quad \Leftrightarrow (1 - \pi _{1}) (T_{1} + c_{2}) + \pi _{1} (T_{1} + c_{1}) \\&\quad \quad > ( 1 - \pi _{1}) (T_{2} + c_{2}) + \pi _{1} \left[ (1 - \eta ) ( T_2 + c_1) + \eta \left( {\frac{p_{2} w}{(1 - p_{2})}} \right) \right] \\&\quad \Leftrightarrow (1-\pi _{1}) (T_{1} - T_{2}) > \pi _{1} \left[ T_{2} + c_{1} - \eta \left( T_{2} + c_{1} - {\frac{p_{2} w}{(1-p_{2})}} \right) - (T_{1} + c_{1}) \right] \\&\quad \Leftrightarrow (1-\pi _{1}) (T_{1} - T_{2}) + \pi _{1} (T_{1} - T_{2}) > -\eta \pi _{1} \left[ T_{2} + c_{1} - {\frac{p_{2} w}{(1-p_2)}} \right] \\&\quad \Leftrightarrow -\eta < {\frac{(T_{1} - T_{2})}{\pi _{1} \left[ T_{2} + c_{1} - {\frac{p_{2} w}{(1-p_2)}} \right] }}\\&\quad \Leftrightarrow \eta ^{*} = {\frac{(T_{2} - T_{1})}{\pi _{1} \left[ T_{2} + c_{1} - {\frac{p_{2} w}{(1-p_{2})}}\right] }} \end{aligned}$$

Hence, $\upeta >\upeta ^{*}$ to have $FC_{sc} > FC_{c}$ $\square$

Proposition 4c

Under a fixed proportion $\pi _1$ of corrupt type 1 bureaucrats, there may be an efficient fiscal policy $(\tau _{2l-w}, p_1)$ that we call lose–win where the government sees its tax revenues decrease and firms see their transfers decrease by reducing the tax rate, $\tau _2$, in order to keep all bureaucrats opportunity cost under $P_{1}$. In such circumstances the tax rate is:

$$\begin{aligned} \tau _{2l-w} < {\frac{p_{1} w}{V(1-p_1)}} - {\frac{c_2}{V}} \end{aligned}$$

Fig. 6

Win–win versus lose–win situation

Proof

Given $(\tau _{2l-w}, p_{1})$, the opportunity cost of both types of bureaucrats is now at $P_{2 l-w}$ in Fig. 6. The Government’s tax revenues from bureaucrat 2 decline from $T_{2w-w}+c_{2}$ to $T_{2l-w}+c_{2}$. However, the situation is to the advantage of firms dealing with type 1 bureaucrats since the lowering of the bureaucrats’ opportunity cost has the effect of lowering the minimal bribe a bureaucrat is willing to accept. In Fig. 6, there is a range (brace $A_{l-w}$) where firms with sufficient bargaining power can lower their bribe to an amount smaller than what they would have paid in fiscal obligations under a no-corruption policy $T_{1}+c_{1}$. We also note that $B_{min \, l-w}$ is effectively smaller than $B_{min \, w-w}= T_{1}+c_{1}$. $\square$

Corollary to Proposition 4c

There is an optimal tax rate $\tau _2^{**}$ set accordingly to $\eta ^{*}$ for which gains made by firms dealing with corrupt type 1 bureaucrats make up for the government’s losses due to the reduction of tax revenues. This optimal tax rate is such that:

$$\begin{aligned} \tau _2^*= {\frac{\left( \tau _{1} + {\frac{\eta ^{*} \pi _{1} c_{1}}{V}} - {\frac{\eta ^{*} \pi _{1} p_{2} w}{V(1-p_2)}} \right) }{(1-\eta ^{*} \pi _{1})}} \end{aligned}$$

Proof

By comparing the effect of the flexible policy and the no-corruption policy on firm cost we obtained a critical bargaining power $\eta ^{*}$. Transforming this equation and isolating $\tau _2^{**}$ yields the optimal tax rate under which firms dealing with corrupt bureaucrats minimize their bribes while the government still collects higher tax revenues with a flexible policy than with a no-corruption policy. $\square$